What is ALGEBRAIC GEOMETRY? What does ALGEBRAIC GEOMETRY mean? ALGEBRAIC GEOMETRY meaning - ALGEBRAIC GEOMETRY definition -ALGEBRAIC GEOMETRY explanation.
Source: Wikipedia.org article, adapted under https://creativecommons.org/licenses/by-sa/3.0/ license.
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.
The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves and quartic curves like lemniscates, and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.
Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.
In the20th century, algebraic geometry split into several subareas.
The mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an algebraically closed field.
The study of the points of an algebraic variety with coordinates in the field of the rational numbers or in a number field became arithmetic geometry (or more classically Diophantine geometry), a subfield of algebraic number theory.
The study of the real points of an algebraic variety is the subject of real algebraic geometry.
A large part of singularity theory is devoted to the singularities of algebraic varieties.
With the rise of the computers, a computational algebraic geometry area has emerged, which lies at the intersection of algebraic geometry and computer algebra. It consists essentially in developing algorithms and software for studying and finding the properties of explicitly given algebraic varieties.
Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology, differential and complex geometry. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory. Wiles's proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach.

published:08 Nov 2016

views:835

In many ways, the modern revival of algebraic geometry resulted from investigations in complex function theory. In this video, we show how Weierstrass's p-function can be used to relate cubic curves and 2-tori. We see in particular that cubic curves have an abelian group structure.
The material in this video and the other in the playlist can be found in Kirwan's "Complex algebraic curves".

The role of curvature in algebraic geometry: some history and current issues
Plática dada por Phillip Griffiths (Institute for Advanced Study, Princeton) en el evento Algebraic Geometry in Mexico 2016 el martes 1 de noviembre del 2016.
Abstract:
Curvature has historically guided some of the most important results in algebraic geometry: vanishing theorems leading to existence results, holomorphic mappings to algebraic varieties (hyperbolicity), and global geometric applications of Hodge theory. Typically curvature methods prove a result when the variety is smooth and the curvature form is definite, and then considerable skill and technical effort extends these results to a general algebro-geometric setting when the variety is singular and the curvature is only definite on a Zariski open set. There has recently been work using curvature properties of the Hodge bundles in a family of algebraic varieties to establish results in the smooth and definite case such as the well-known conjectures “The Hodge bundle is ample on the Satake-Baily-Borel completion of a period mapping” and “The moduli space for varieties of general type is itself of log-general type”. The proofs of these results in general require non-trivial technical extensions in the directions suggested by curvature considerations, including extending the methods to include when the metric and its curvature become singular. In this talk I will briefly recall some of the classical results and then discuss some more recent results and issues just mentioned.
Video de la plática
https://youtu.be/ceBhyAeKRrs
Foto de miniatura del video
https://goo.gl/photos/A5ZVZ2sGxFpqek9x9
Página de Phillip Griffiths en Wikipedia
https://en.wikipedia.org/wiki/Phillip_Griffiths
Página de Phillip Griffiths en Phillip Griffiths en Institute for Advanced Study, Princeton
https://www.ias.edu/scholars/griffiths
Otras pláticas del evento Algebraic Geometry in Mexico 2016
https://www.youtube.com/playlist?list=PLrg-5oUhFeioieJgNVvON2WGwIh07bh8M
Otras pláticas del evento Algebraic Geometry in Mexico 2015
https://www.youtube.com/playlist?list=PLrg-5oUhFeipLL7JxLlnij0z1EKQ1dr6i
Videos Académicos de matemáticas (pláticas grabadas)
https://www.youtube.com/playlist?list=PLrg-5oUhFeiqvXI_Bxb5iOH25gXprez6o
Videos Académicos publicados
https://www.youtube.com/playlist?list=PLrg-5oUhFeirZBmxs7kDn_KrKHUJradjz
Videos Académicos publicados en 2016
https://www.youtube.com/playlist?list=PLrg-5oUhFeirKKFi4xM3LtzuCbHlKlMJO
Album de fotos miniatura de los videos Académicos publicados en 2016
https://goo.gl/photos/Wk45yoGqPe5hKTw77
Videos Académicos publicados en 2015
https://www.youtube.com/playlist?list=PLrg-5oUhFeiosqyOgMN2yPybqXIrPAhbi
Videos Académicos publicados en 2014
https://www.youtube.com/playlist?list=PLrg-5oUhFeiomqff3PibNQMJJngwB-wL6
Videos Académicos publicados en 2013
https://www.youtube.com/playlist?list=PLrg-5oUhFeipDUZKuJZM0QJDP22Cogkot
Agradecemos el apoyo de
universo.math​
http://universo.math.org.mx/
https://www.facebook.com/universo.math
Departamento de Matemáticas del CINVESTAV
http://www.math.cinvestav.mx/
Facultad de Ciencias de la UNAM
http://www.fciencias.unam.mx/
https://www.facebook.com/Facultad-de-Ciencias-214278861928417/?fref=ts

This is the first in a series of videos introducing affine algebraic geometry. One of the principles of algebraic geometry is that the co-ordinate ring captures all the geometry. In this video, we show how points of affine varieties can be described purely algebraically in terms of the co-ordinate ring. Hence the co-ordinate ring recovers the underlying set structure of an affine variety.
The material in this video and the entire playlist is standard and can be found in many introductory texts on algebraic geometry including Shafarevich's "BasicAlgebraic Geometry" and Hartshorne's "Algebraic Geometry".

published:09 Mar 2017

views:719

published:29 Jul 2014

views:20676

This video is a tutorial on Algebra & GeometryQuestions. You should have already watched the Algebra & Geometry 1 Tutorial 25. This video is for students attempting the Higher paper AQAUnit 3 MathsGCSE, who have previously sat the foundation paper. Explanations are aimed at being as simple as possible and so students who previously did the Foundation paper can access. They would also be useful for students who have always sat the higher paper. www.hegartymaths.com http://www.hegartymaths.com/

Geometry

Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales (6th century BC). By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. In the classical world, both geometry and astronomy were considered to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.

Algebra

Algebra (from Arabic"al-jabr" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the Arabic origin of its name suggests, was done in the Middle East, by mathematicians such as al-Khwārizmī (780 – 850) and Omar Khayyam (1048–1131).

Algebraic Geometry - Lothar Göttsche - Lecture 01

What is ALGEBRAIC GEOMETRY? What does ALGEBRAIC GEOMETRY mean? ALGEBRAIC GEOMETRY meaning

What is ALGEBRAIC GEOMETRY? What does ALGEBRAIC GEOMETRY mean? ALGEBRAIC GEOMETRY meaning

What is ALGEBRAIC GEOMETRY? What does ALGEBRAIC GEOMETRY mean? ALGEBRAIC GEOMETRY meaning

What is ALGEBRAIC GEOMETRY? What does ALGEBRAIC GEOMETRY mean? ALGEBRAIC GEOMETRY meaning - ALGEBRAIC GEOMETRY definition -ALGEBRAIC GEOMETRY explanation.
Source: Wikipedia.org article, adapted under https://creativecommons.org/licenses/by-sa/3.0/ license.
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.
The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves and quartic curves like lemniscates, and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.
Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.
In the20th century, algebraic geometry split into several subareas.
The mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an algebraically closed field.
The study of the points of an algebraic variety with coordinates in the field of the rational numbers or in a number field became arithmetic geometry (or more classically Diophantine geometry), a subfield of algebraic number theory.
The study of the real points of an algebraic variety is the subject of real algebraic geometry.
A large part of singularity theory is devoted to the singularities of algebraic varieties.
With the rise of the computers, a computational algebraic geometry area has emerged, which lies at the intersection of algebraic geometry and computer algebra. It consists essentially in developing algorithms and software for studying and finding the properties of explicitly given algebraic varieties.
Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology, differential and complex geometry. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory. Wiles's proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach.

18:57

Complex Analysis and algebraic geometry

Complex Analysis and algebraic geometry

Complex Analysis and algebraic geometry

In many ways, the modern revival of algebraic geometry resulted from investigations in complex function theory. In this video, we show how Weierstrass's p-function can be used to relate cubic curves and 2-tori. We see in particular that cubic curves have an abelian group structure.
The material in this video and the other in the playlist can be found in Kirwan's "Complex algebraic curves".

The role of curvature in algebraic geometry: some history and current issues (Phillip Griffiths)

The role of curvature in algebraic geometry: some history and current issues (Phillip Griffiths)

The role of curvature in algebraic geometry: some history and current issues (Phillip Griffiths)

The role of curvature in algebraic geometry: some history and current issues
Plática dada por Phillip Griffiths (Institute for Advanced Study, Princeton) en el evento Algebraic Geometry in Mexico 2016 el martes 1 de noviembre del 2016.
Abstract:
Curvature has historically guided some of the most important results in algebraic geometry: vanishing theorems leading to existence results, holomorphic mappings to algebraic varieties (hyperbolicity), and global geometric applications of Hodge theory. Typically curvature methods prove a result when the variety is smooth and the curvature form is definite, and then considerable skill and technical effort extends these results to a general algebro-geometric setting when the variety is singular and the curvature is only definite on a Zariski open set. There has recently been work using curvature properties of the Hodge bundles in a family of algebraic varieties to establish results in the smooth and definite case such as the well-known conjectures “The Hodge bundle is ample on the Satake-Baily-Borel completion of a period mapping” and “The moduli space for varieties of general type is itself of log-general type”. The proofs of these results in general require non-trivial technical extensions in the directions suggested by curvature considerations, including extending the methods to include when the metric and its curvature become singular. In this talk I will briefly recall some of the classical results and then discuss some more recent results and issues just mentioned.
Video de la plática
https://youtu.be/ceBhyAeKRrs
Foto de miniatura del video
https://goo.gl/photos/A5ZVZ2sGxFpqek9x9
Página de Phillip Griffiths en Wikipedia
https://en.wikipedia.org/wiki/Phillip_Griffiths
Página de Phillip Griffiths en Phillip Griffiths en Institute for Advanced Study, Princeton
https://www.ias.edu/scholars/griffiths
Otras pláticas del evento Algebraic Geometry in Mexico 2016
https://www.youtube.com/playlist?list=PLrg-5oUhFeioieJgNVvON2WGwIh07bh8M
Otras pláticas del evento Algebraic Geometry in Mexico 2015
https://www.youtube.com/playlist?list=PLrg-5oUhFeipLL7JxLlnij0z1EKQ1dr6i
Videos Académicos de matemáticas (pláticas grabadas)
https://www.youtube.com/playlist?list=PLrg-5oUhFeiqvXI_Bxb5iOH25gXprez6o
Videos Académicos publicados
https://www.youtube.com/playlist?list=PLrg-5oUhFeirZBmxs7kDn_KrKHUJradjz
Videos Académicos publicados en 2016
https://www.youtube.com/playlist?list=PLrg-5oUhFeirKKFi4xM3LtzuCbHlKlMJO
Album de fotos miniatura de los videos Académicos publicados en 2016
https://goo.gl/photos/Wk45yoGqPe5hKTw77
Videos Académicos publicados en 2015
https://www.youtube.com/playlist?list=PLrg-5oUhFeiosqyOgMN2yPybqXIrPAhbi
Videos Académicos publicados en 2014
https://www.youtube.com/playlist?list=PLrg-5oUhFeiomqff3PibNQMJJngwB-wL6
Videos Académicos publicados en 2013
https://www.youtube.com/playlist?list=PLrg-5oUhFeipDUZKuJZM0QJDP22Cogkot
Agradecemos el apoyo de
universo.math​
http://universo.math.org.mx/
https://www.facebook.com/universo.math
Departamento de Matemáticas del CINVESTAV
http://www.math.cinvestav.mx/
Facultad de Ciencias de la UNAM
http://www.fciencias.unam.mx/
https://www.facebook.com/Facultad-de-Ciencias-214278861928417/?fref=ts

Affine algebraic geometry: Algebraic Incarnation of Points

This is the first in a series of videos introducing affine algebraic geometry. One of the principles of algebraic geometry is that the co-ordinate ring captures all the geometry. In this video, we show how points of affine varieties can be described purely algebraically in terms of the co-ordinate ring. Hence the co-ordinate ring recovers the underlying set structure of an affine variety.
The material in this video and the entire playlist is standard and can be found in many introductory texts on algebraic geometry including Shafarevich's "BasicAlgebraic Geometry" and Hartshorne's "Algebraic Geometry".

13:07

1 Intro to Algebraic Geometry

1 Intro to Algebraic Geometry

1 Intro to Algebraic Geometry

32:05

Algebra & Geometry 1 (GCSE Higher Maths) - Exam Qs 26

Algebra & Geometry 1 (GCSE Higher Maths) - Exam Qs 26

Algebra & Geometry 1 (GCSE Higher Maths) - Exam Qs 26

This video is a tutorial on Algebra & GeometryQuestions. You should have already watched the Algebra & Geometry 1 Tutorial 25. This video is for students attempting the Higher paper AQAUnit 3 MathsGCSE, who have previously sat the foundation paper. Explanations are aimed at being as simple as possible and so students who previously did the Foundation paper can access. They would also be useful for students who have always sat the higher paper. www.hegartymaths.com http://www.hegartymaths.com/

Week 10 - Algebraic and Geometric Multiplicities of an Eigenvalue

3:18

Geometry Proofs - Algebra Proofs - MathHelp.com

Geometry Proofs - Algebra Proofs - MathHelp.com

Geometry Proofs - Algebra Proofs - MathHelp.com

For complete lessons on geometry proofs and algebra proofs, go to http://www.MathHelp.com - 1000+ online math lessons featuring a personal math teacher inside every lesson! In this lesson, students are asked to provide the missing reasons in two-column Algebra proofs using the properties of equality.

Algebraic Geometry - Lothar Göttsche - Lecture 01

published: 09 Dec 2016

What is ALGEBRAIC GEOMETRY? What does ALGEBRAIC GEOMETRY mean? ALGEBRAIC GEOMETRY meaning

What is ALGEBRAIC GEOMETRY? What does ALGEBRAIC GEOMETRY mean? ALGEBRAIC GEOMETRY meaning - ALGEBRAIC GEOMETRY definition -ALGEBRAIC GEOMETRY explanation.
Source: Wikipedia.org article, adapted under https://creativecommons.org/licenses/by-sa/3.0/ license.
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.
The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which inclu...

published: 08 Nov 2016

Complex Analysis and algebraic geometry

In many ways, the modern revival of algebraic geometry resulted from investigations in complex function theory. In this video, we show how Weierstrass's p-function can be used to relate cubic curves and 2-tori. We see in particular that cubic curves have an abelian group structure.
The material in this video and the other in the playlist can be found in Kirwan's "Complex algebraic curves".

Algebra: Formulas From Geometry

This lesson consists of providing you with a basic review of the formulas from geometry you will most likely encounter in algebra (and other math classes like trigonometry and calculus). This is NOT a detailed, formal lesson.
Formulas and concepts covered include:
Angles (complementary & supplementary)
Triangles (perimeter and area)
Isosceles triangleEquilateral triangleRight triangle
The Pythagorean TheoremSimilar trianglesPerimeter and area formulas for Quadrilaterals (square, rectangle, parallelogram, trapezoid). Some diagonal formulas too, where applicable.
Circles (radius, diameter, circumference, area)
Cube (diagonal of face, diagonal of cube, surface area and volume)
Rectangular box [or Rectangular Parallelepiped] (diagonal of box, surface area and volume)
Prism...

The role of curvature in algebraic geometry: some history and current issues (Phillip Griffiths)

The role of curvature in algebraic geometry: some history and current issues
Plática dada por Phillip Griffiths (Institute for Advanced Study, Princeton) en el evento Algebraic Geometry in Mexico 2016 el martes 1 de noviembre del 2016.
Abstract:
Curvature has historically guided some of the most important results in algebraic geometry: vanishing theorems leading to existence results, holomorphic mappings to algebraic varieties (hyperbolicity), and global geometric applications of Hodge theory. Typically curvature methods prove a result when the variety is smooth and the curvature form is definite, and then considerable skill and technical effort extends these results to a general algebro-geometric setting when the variety is singular and the curvature is only definite on a Zariski open...

Affine algebraic geometry: Algebraic Incarnation of Points

This is the first in a series of videos introducing affine algebraic geometry. One of the principles of algebraic geometry is that the co-ordinate ring captures all the geometry. In this video, we show how points of affine varieties can be described purely algebraically in terms of the co-ordinate ring. Hence the co-ordinate ring recovers the underlying set structure of an affine variety.
The material in this video and the entire playlist is standard and can be found in many introductory texts on algebraic geometry including Shafarevich's "BasicAlgebraic Geometry" and Hartshorne's "Algebraic Geometry".

published: 09 Mar 2017

1 Intro to Algebraic Geometry

published: 29 Jul 2014

Algebra & Geometry 1 (GCSE Higher Maths) - Exam Qs 26

This video is a tutorial on Algebra & GeometryQuestions. You should have already watched the Algebra & Geometry 1 Tutorial 25. This video is for students attempting the Higher paper AQAUnit 3 MathsGCSE, who have previously sat the foundation paper. Explanations are aimed at being as simple as possible and so students who previously did the Foundation paper can access. They would also be useful for students who have always sat the higher paper. www.hegartymaths.com http://www.hegartymaths.com/

This online sat math test prep review youtube video tutorial will help you to learn the fundamentals behind the main concepts that are routinely covered on the scholastic aptitude test. This online crash course video contains plenty of examples and practice problems for you work on including very hard / difficult math questions with answers and solutions included. There are six main lessons in this study guide that are accompanied by a review of the most important topics, concepts, equations and formulas that you need to do well on the sat. This video contains plenty of multiple choice problems that you can work on as a practice test.
Extended SATVideo: https://vimeo.com/ondemand/satmathreview
AlgebraOnlineCourse:
https://www.udemy.com/algebracourse7245/learn/v4/content
Access to...

published: 16 Jun 2016

Week 10 - Algebraic and Geometric Multiplicities of an Eigenvalue

published: 10 Dec 2013

Geometry Proofs - Algebra Proofs - MathHelp.com

For complete lessons on geometry proofs and algebra proofs, go to http://www.MathHelp.com - 1000+ online math lessons featuring a personal math teacher inside every lesson! In this lesson, students are asked to provide the missing reasons in two-column Algebra proofs using the properties of equality.

What is ALGEBRAIC GEOMETRY? What does ALGEBRAIC GEOMETRY mean? ALGEBRAIC GEOMETRY meaning - ALGEBRAIC GEOMETRY definition -ALGEBRAIC GEOMETRY explanation.
Source: Wikipedia.org article, adapted under https://creativecommons.org/licenses/by-sa/3.0/ license.
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.
The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves and quartic curves like lemniscates, and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.
Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.
In the20th century, algebraic geometry split into several subareas.
The mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an algebraically closed field.
The study of the points of an algebraic variety with coordinates in the field of the rational numbers or in a number field became arithmetic geometry (or more classically Diophantine geometry), a subfield of algebraic number theory.
The study of the real points of an algebraic variety is the subject of real algebraic geometry.
A large part of singularity theory is devoted to the singularities of algebraic varieties.
With the rise of the computers, a computational algebraic geometry area has emerged, which lies at the intersection of algebraic geometry and computer algebra. It consists essentially in developing algorithms and software for studying and finding the properties of explicitly given algebraic varieties.
Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology, differential and complex geometry. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory. Wiles's proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach.

What is ALGEBRAIC GEOMETRY? What does ALGEBRAIC GEOMETRY mean? ALGEBRAIC GEOMETRY meaning - ALGEBRAIC GEOMETRY definition -ALGEBRAIC GEOMETRY explanation.
Source: Wikipedia.org article, adapted under https://creativecommons.org/licenses/by-sa/3.0/ license.
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.
The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves and quartic curves like lemniscates, and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.
Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.
In the20th century, algebraic geometry split into several subareas.
The mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an algebraically closed field.
The study of the points of an algebraic variety with coordinates in the field of the rational numbers or in a number field became arithmetic geometry (or more classically Diophantine geometry), a subfield of algebraic number theory.
The study of the real points of an algebraic variety is the subject of real algebraic geometry.
A large part of singularity theory is devoted to the singularities of algebraic varieties.
With the rise of the computers, a computational algebraic geometry area has emerged, which lies at the intersection of algebraic geometry and computer algebra. It consists essentially in developing algorithms and software for studying and finding the properties of explicitly given algebraic varieties.
Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology, differential and complex geometry. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory. Wiles's proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach.

Complex Analysis and algebraic geometry

In many ways, the modern revival of algebraic geometry resulted from investigations in complex function theory. In this video, we show how Weierstrass's p-funct...

In many ways, the modern revival of algebraic geometry resulted from investigations in complex function theory. In this video, we show how Weierstrass's p-function can be used to relate cubic curves and 2-tori. We see in particular that cubic curves have an abelian group structure.
The material in this video and the other in the playlist can be found in Kirwan's "Complex algebraic curves".

In many ways, the modern revival of algebraic geometry resulted from investigations in complex function theory. In this video, we show how Weierstrass's p-function can be used to relate cubic curves and 2-tori. We see in particular that cubic curves have an abelian group structure.
The material in this video and the other in the playlist can be found in Kirwan's "Complex algebraic curves".

The role of curvature in algebraic geometry: some history and current issues
Plática dada por Phillip Griffiths (Institute for Advanced Study, Princeton) en el evento Algebraic Geometry in Mexico 2016 el martes 1 de noviembre del 2016.
Abstract:
Curvature has historically guided some of the most important results in algebraic geometry: vanishing theorems leading to existence results, holomorphic mappings to algebraic varieties (hyperbolicity), and global geometric applications of Hodge theory. Typically curvature methods prove a result when the variety is smooth and the curvature form is definite, and then considerable skill and technical effort extends these results to a general algebro-geometric setting when the variety is singular and the curvature is only definite on a Zariski open set. There has recently been work using curvature properties of the Hodge bundles in a family of algebraic varieties to establish results in the smooth and definite case such as the well-known conjectures “The Hodge bundle is ample on the Satake-Baily-Borel completion of a period mapping” and “The moduli space for varieties of general type is itself of log-general type”. The proofs of these results in general require non-trivial technical extensions in the directions suggested by curvature considerations, including extending the methods to include when the metric and its curvature become singular. In this talk I will briefly recall some of the classical results and then discuss some more recent results and issues just mentioned.
Video de la plática
https://youtu.be/ceBhyAeKRrs
Foto de miniatura del video
https://goo.gl/photos/A5ZVZ2sGxFpqek9x9
Página de Phillip Griffiths en Wikipedia
https://en.wikipedia.org/wiki/Phillip_Griffiths
Página de Phillip Griffiths en Phillip Griffiths en Institute for Advanced Study, Princeton
https://www.ias.edu/scholars/griffiths
Otras pláticas del evento Algebraic Geometry in Mexico 2016
https://www.youtube.com/playlist?list=PLrg-5oUhFeioieJgNVvON2WGwIh07bh8M
Otras pláticas del evento Algebraic Geometry in Mexico 2015
https://www.youtube.com/playlist?list=PLrg-5oUhFeipLL7JxLlnij0z1EKQ1dr6i
Videos Académicos de matemáticas (pláticas grabadas)
https://www.youtube.com/playlist?list=PLrg-5oUhFeiqvXI_Bxb5iOH25gXprez6o
Videos Académicos publicados
https://www.youtube.com/playlist?list=PLrg-5oUhFeirZBmxs7kDn_KrKHUJradjz
Videos Académicos publicados en 2016
https://www.youtube.com/playlist?list=PLrg-5oUhFeirKKFi4xM3LtzuCbHlKlMJO
Album de fotos miniatura de los videos Académicos publicados en 2016
https://goo.gl/photos/Wk45yoGqPe5hKTw77
Videos Académicos publicados en 2015
https://www.youtube.com/playlist?list=PLrg-5oUhFeiosqyOgMN2yPybqXIrPAhbi
Videos Académicos publicados en 2014
https://www.youtube.com/playlist?list=PLrg-5oUhFeiomqff3PibNQMJJngwB-wL6
Videos Académicos publicados en 2013
https://www.youtube.com/playlist?list=PLrg-5oUhFeipDUZKuJZM0QJDP22Cogkot
Agradecemos el apoyo de
universo.math​
http://universo.math.org.mx/
https://www.facebook.com/universo.math
Departamento de Matemáticas del CINVESTAV
http://www.math.cinvestav.mx/
Facultad de Ciencias de la UNAM
http://www.fciencias.unam.mx/
https://www.facebook.com/Facultad-de-Ciencias-214278861928417/?fref=ts

The role of curvature in algebraic geometry: some history and current issues
Plática dada por Phillip Griffiths (Institute for Advanced Study, Princeton) en el evento Algebraic Geometry in Mexico 2016 el martes 1 de noviembre del 2016.
Abstract:
Curvature has historically guided some of the most important results in algebraic geometry: vanishing theorems leading to existence results, holomorphic mappings to algebraic varieties (hyperbolicity), and global geometric applications of Hodge theory. Typically curvature methods prove a result when the variety is smooth and the curvature form is definite, and then considerable skill and technical effort extends these results to a general algebro-geometric setting when the variety is singular and the curvature is only definite on a Zariski open set. There has recently been work using curvature properties of the Hodge bundles in a family of algebraic varieties to establish results in the smooth and definite case such as the well-known conjectures “The Hodge bundle is ample on the Satake-Baily-Borel completion of a period mapping” and “The moduli space for varieties of general type is itself of log-general type”. The proofs of these results in general require non-trivial technical extensions in the directions suggested by curvature considerations, including extending the methods to include when the metric and its curvature become singular. In this talk I will briefly recall some of the classical results and then discuss some more recent results and issues just mentioned.
Video de la plática
https://youtu.be/ceBhyAeKRrs
Foto de miniatura del video
https://goo.gl/photos/A5ZVZ2sGxFpqek9x9
Página de Phillip Griffiths en Wikipedia
https://en.wikipedia.org/wiki/Phillip_Griffiths
Página de Phillip Griffiths en Phillip Griffiths en Institute for Advanced Study, Princeton
https://www.ias.edu/scholars/griffiths
Otras pláticas del evento Algebraic Geometry in Mexico 2016
https://www.youtube.com/playlist?list=PLrg-5oUhFeioieJgNVvON2WGwIh07bh8M
Otras pláticas del evento Algebraic Geometry in Mexico 2015
https://www.youtube.com/playlist?list=PLrg-5oUhFeipLL7JxLlnij0z1EKQ1dr6i
Videos Académicos de matemáticas (pláticas grabadas)
https://www.youtube.com/playlist?list=PLrg-5oUhFeiqvXI_Bxb5iOH25gXprez6o
Videos Académicos publicados
https://www.youtube.com/playlist?list=PLrg-5oUhFeirZBmxs7kDn_KrKHUJradjz
Videos Académicos publicados en 2016
https://www.youtube.com/playlist?list=PLrg-5oUhFeirKKFi4xM3LtzuCbHlKlMJO
Album de fotos miniatura de los videos Académicos publicados en 2016
https://goo.gl/photos/Wk45yoGqPe5hKTw77
Videos Académicos publicados en 2015
https://www.youtube.com/playlist?list=PLrg-5oUhFeiosqyOgMN2yPybqXIrPAhbi
Videos Académicos publicados en 2014
https://www.youtube.com/playlist?list=PLrg-5oUhFeiomqff3PibNQMJJngwB-wL6
Videos Académicos publicados en 2013
https://www.youtube.com/playlist?list=PLrg-5oUhFeipDUZKuJZM0QJDP22Cogkot
Agradecemos el apoyo de
universo.math​
http://universo.math.org.mx/
https://www.facebook.com/universo.math
Departamento de Matemáticas del CINVESTAV
http://www.math.cinvestav.mx/
Facultad de Ciencias de la UNAM
http://www.fciencias.unam.mx/
https://www.facebook.com/Facultad-de-Ciencias-214278861928417/?fref=ts

Affine algebraic geometry: Algebraic Incarnation of Points

This is the first in a series of videos introducing affine algebraic geometry. One of the principles of algebraic geometry is that the co-ordinate ring captures...

This is the first in a series of videos introducing affine algebraic geometry. One of the principles of algebraic geometry is that the co-ordinate ring captures all the geometry. In this video, we show how points of affine varieties can be described purely algebraically in terms of the co-ordinate ring. Hence the co-ordinate ring recovers the underlying set structure of an affine variety.
The material in this video and the entire playlist is standard and can be found in many introductory texts on algebraic geometry including Shafarevich's "BasicAlgebraic Geometry" and Hartshorne's "Algebraic Geometry".

This is the first in a series of videos introducing affine algebraic geometry. One of the principles of algebraic geometry is that the co-ordinate ring captures all the geometry. In this video, we show how points of affine varieties can be described purely algebraically in terms of the co-ordinate ring. Hence the co-ordinate ring recovers the underlying set structure of an affine variety.
The material in this video and the entire playlist is standard and can be found in many introductory texts on algebraic geometry including Shafarevich's "BasicAlgebraic Geometry" and Hartshorne's "Algebraic Geometry".

Algebra & Geometry 1 (GCSE Higher Maths) - Exam Qs 26

This video is a tutorial on Algebra & GeometryQuestions. You should have already watched the Algebra & Geometry 1 Tutorial 25. This video is for students atte...

This video is a tutorial on Algebra & GeometryQuestions. You should have already watched the Algebra & Geometry 1 Tutorial 25. This video is for students attempting the Higher paper AQAUnit 3 MathsGCSE, who have previously sat the foundation paper. Explanations are aimed at being as simple as possible and so students who previously did the Foundation paper can access. They would also be useful for students who have always sat the higher paper. www.hegartymaths.com http://www.hegartymaths.com/

This video is a tutorial on Algebra & GeometryQuestions. You should have already watched the Algebra & Geometry 1 Tutorial 25. This video is for students attempting the Higher paper AQAUnit 3 MathsGCSE, who have previously sat the foundation paper. Explanations are aimed at being as simple as possible and so students who previously did the Foundation paper can access. They would also be useful for students who have always sat the higher paper. www.hegartymaths.com http://www.hegartymaths.com/

For complete lessons on geometry proofs and algebra proofs, go to http://www.MathHelp.com - 1000+ online math lessons featuring a personal math teacher inside every lesson! In this lesson, students are asked to provide the missing reasons in two-column Algebra proofs using the properties of equality.

For complete lessons on geometry proofs and algebra proofs, go to http://www.MathHelp.com - 1000+ online math lessons featuring a personal math teacher inside every lesson! In this lesson, students are asked to provide the missing reasons in two-column Algebra proofs using the properties of equality.

Prof. Jean Dieudonné: "The Historical Development of Algebraic Geometry"

Algebra: Formulas From Geometry

This lesson consists of providing you with a basic review of the formulas from geometry you will most likely encounter in algebra (and other math classes like trigonometry and calculus). This is NOT a detailed, formal lesson.
Formulas and concepts covered include:
Angles (complementary & supplementary)
Triangles (perimeter and area)
Isosceles triangleEquilateral triangleRight triangle
The Pythagorean TheoremSimilar trianglesPerimeter and area formulas for Quadrilaterals (square, rectangle, parallelogram, trapezoid). Some diagonal formulas too, where applicable.
Circles (radius, diameter, circumference, area)
Cube (diagonal of face, diagonal of cube, surface area and volume)
Rectangular box [or Rectangular Parallelepiped] (diagonal of box, surface area and volume)
Prism...

published: 26 Nov 2008

Euclidean and Algebraic Geometry, David Cox [2014]

slides for this talk: http://helper.ipam.ucla.edu/publications/ccgws1/ccgws1_11873.pdfDavid CoxAmherst College
This talk will survey some examples, mostly geometric questions about Euclidean space, where the methods of algebraic geometry can offer some insight. I will also discuss some of the limitations of this approach.
http://www.ipam.ucla.edu/abstract/?tid=11873&pcode=CCGWS1

From the MSRI introductory workshop on "Algebraic Topology" that took place at MSRI, Berkeley in January 2014. The workshop page (with videos and references and exercises) can be found here:
http://www.msri.org/workshops/685
The page for this particular talk is:
http://www.msri.org/workshops/685/schedules/17910

published: 06 Aug 2014

Algebra & Geometry 1 (GCSE Higher Maths) - Exam Qs 26

This video is a tutorial on Algebra & GeometryQuestions. You should have already watched the Algebra & Geometry 1 Tutorial 25. This video is for students attempting the Higher paper AQAUnit 3 MathsGCSE, who have previously sat the foundation paper. Explanations are aimed at being as simple as possible and so students who previously did the Foundation paper can access. They would also be useful for students who have always sat the higher paper. www.hegartymaths.com http://www.hegartymaths.com/

Abstract:
We will briefly recall the general philosophy of non-commutative (and derived) algebraic geometry in order to establish a precise link between dg-derived category of singularities of Landau-Ginzburg models and vanishing cohomology, over an arbitrary henselian trait. We will then focus on a trace formula for dg-categories and a recent application to Bloch’s conductor conjecture. This second, and main part of the talk refers to work in progress, joint with B. Toën.

An accessible lecture by Jacob Lurie from about 2005.
The motivations behind derived algebraic geometry are presented, starting with Bezout's theorem for intersections.
A generalization of algebraic geometry using derived (topological) coordinate rings -- where identifications (quotients by ideals of defining equations) are carried out "topologically" -- allows for the proper handling of non-transversal intersections, justifying Serre's otherwise mysterious formula.
See also the paper:
http://www.math.harvard.edu/~lurie/papers/DAG.pdf
And the book:
http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf

This online sat math test prep review youtube video tutorial will help you to learn the fundamentals behind the main concepts that are routinely covered on the scholastic aptitude test. This online crash course video contains plenty of examples and practice problems for you work on including very hard / difficult math questions with answers and solutions included. There are six main lessons in this study guide that are accompanied by a review of the most important topics, concepts, equations and formulas that you need to do well on the sat. This video contains plenty of multiple choice problems that you can work on as a practice test.
Extended SATVideo: https://vimeo.com/ondemand/satmathreview
AlgebraOnlineCourse:
https://www.udemy.com/algebracourse7245/learn/v4/content
Access to...

Algebraic geometry seminar
Department of Pure MathematicsUniversity of Waterloo
July 27th, 2017
Abstract:
We will continue with the discussion about morphisms of schemes. We will see an explicit example of constructing morphisms of schemes by cutting up into affine open sets and then gluing them together. Then, we will define Z-valued points, where Z is a given scheme. This notion will lead us to determine a morphism not by its values at points but rather by its induced map of Z-valued points.
Following the notes of Ravi Vakil, available at http://math.stanford.edu/~vakil/216blog/index.html.

From the MSRI workshop "Reimagining the Foundations of Algebraic Topology" that took place at MSRI, Berkeley in April 2014. The workshop page (with videos and some supplemental material) can be found here:
http://www.msri.org/workshops/689
The page for this particular talk is:
http://www.msri.org/workshops/689/schedules/18216

This is a gentle introduction to curves and more specifically algebraic curves. We look at historical aspects of curves, going back to the ancient Greeks, then on the 17th century work of Descartes.
We point out some of the difficulties with Jordan's notion of curve, and move to the polynumber approach to algebraic curves.
The aim is to set the stage to generalize the algebraic calculus of the previous few lectures to algebraic curves.
This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. The full playlist is at http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=view_all
A screenshot PDF which includes MathFound...

slides for this talk: http://helper.ipam.ucla.edu/publications/ccgws1/ccgws1_11873.pdfDavid CoxAmherst College
This talk will survey some examples, mostly geometric questions about Euclidean space, where the methods of algebraic geometry can offer some insight. I will also discuss some of the limitations of this approach.
http://www.ipam.ucla.edu/abstract/?tid=11873&pcode=CCGWS1

slides for this talk: http://helper.ipam.ucla.edu/publications/ccgws1/ccgws1_11873.pdfDavid CoxAmherst College
This talk will survey some examples, mostly geometric questions about Euclidean space, where the methods of algebraic geometry can offer some insight. I will also discuss some of the limitations of this approach.
http://www.ipam.ucla.edu/abstract/?tid=11873&pcode=CCGWS1

From the MSRI introductory workshop on "Algebraic Topology" that took place at MSRI, Berkeley in January 2014. The workshop page (with videos and references and exercises) can be found here:
http://www.msri.org/workshops/685
The page for this particular talk is:
http://www.msri.org/workshops/685/schedules/17910

From the MSRI introductory workshop on "Algebraic Topology" that took place at MSRI, Berkeley in January 2014. The workshop page (with videos and references and exercises) can be found here:
http://www.msri.org/workshops/685
The page for this particular talk is:
http://www.msri.org/workshops/685/schedules/17910

Algebra & Geometry 1 (GCSE Higher Maths) - Exam Qs 26

This video is a tutorial on Algebra & GeometryQuestions. You should have already watched the Algebra & Geometry 1 Tutorial 25. This video is for students atte...

This video is a tutorial on Algebra & GeometryQuestions. You should have already watched the Algebra & Geometry 1 Tutorial 25. This video is for students attempting the Higher paper AQAUnit 3 MathsGCSE, who have previously sat the foundation paper. Explanations are aimed at being as simple as possible and so students who previously did the Foundation paper can access. They would also be useful for students who have always sat the higher paper. www.hegartymaths.com http://www.hegartymaths.com/

This video is a tutorial on Algebra & GeometryQuestions. You should have already watched the Algebra & Geometry 1 Tutorial 25. This video is for students attempting the Higher paper AQAUnit 3 MathsGCSE, who have previously sat the foundation paper. Explanations are aimed at being as simple as possible and so students who previously did the Foundation paper can access. They would also be useful for students who have always sat the higher paper. www.hegartymaths.com http://www.hegartymaths.com/

Abstract:
We will briefly recall the general philosophy of non-commutative (and derived) algebraic geometry in order to establish a precise link between dg-deri...

Abstract:
We will briefly recall the general philosophy of non-commutative (and derived) algebraic geometry in order to establish a precise link between dg-derived category of singularities of Landau-Ginzburg models and vanishing cohomology, over an arbitrary henselian trait. We will then focus on a trace formula for dg-categories and a recent application to Bloch’s conductor conjecture. This second, and main part of the talk refers to work in progress, joint with B. Toën.

Abstract:
We will briefly recall the general philosophy of non-commutative (and derived) algebraic geometry in order to establish a precise link between dg-derived category of singularities of Landau-Ginzburg models and vanishing cohomology, over an arbitrary henselian trait. We will then focus on a trace formula for dg-categories and a recent application to Bloch’s conductor conjecture. This second, and main part of the talk refers to work in progress, joint with B. Toën.

An accessible lecture by Jacob Lurie from about 2005.
The motivations behind derived algebraic geometry are presented, starting with Bezout's theorem for intersections.
A generalization of algebraic geometry using derived (topological) coordinate rings -- where identifications (quotients by ideals of defining equations) are carried out "topologically" -- allows for the proper handling of non-transversal intersections, justifying Serre's otherwise mysterious formula.
See also the paper:
http://www.math.harvard.edu/~lurie/papers/DAG.pdf
And the book:
http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf

An accessible lecture by Jacob Lurie from about 2005.
The motivations behind derived algebraic geometry are presented, starting with Bezout's theorem for intersections.
A generalization of algebraic geometry using derived (topological) coordinate rings -- where identifications (quotients by ideals of defining equations) are carried out "topologically" -- allows for the proper handling of non-transversal intersections, justifying Serre's otherwise mysterious formula.
See also the paper:
http://www.math.harvard.edu/~lurie/papers/DAG.pdf
And the book:
http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf

Algebraic geometry seminar
Department of Pure MathematicsUniversity of Waterloo
July 27th, 2017
Abstract:
We will continue with the discussion about morphisms of schemes. We will see an explicit example of constructing morphisms of schemes by cutting up into affine open sets and then gluing them together. Then, we will define Z-valued points, where Z is a given scheme. This notion will lead us to determine a morphism not by its values at points but rather by its induced map of Z-valued points.
Following the notes of Ravi Vakil, available at http://math.stanford.edu/~vakil/216blog/index.html.

Algebraic geometry seminar
Department of Pure MathematicsUniversity of Waterloo
July 27th, 2017
Abstract:
We will continue with the discussion about morphisms of schemes. We will see an explicit example of constructing morphisms of schemes by cutting up into affine open sets and then gluing them together. Then, we will define Z-valued points, where Z is a given scheme. This notion will lead us to determine a morphism not by its values at points but rather by its induced map of Z-valued points.
Following the notes of Ravi Vakil, available at http://math.stanford.edu/~vakil/216blog/index.html.

From the MSRI workshop "Reimagining the Foundations of Algebraic Topology" that took place at MSRI, Berkeley in April 2014. The workshop page (with videos and some supplemental material) can be found here:
http://www.msri.org/workshops/689
The page for this particular talk is:
http://www.msri.org/workshops/689/schedules/18216

From the MSRI workshop "Reimagining the Foundations of Algebraic Topology" that took place at MSRI, Berkeley in April 2014. The workshop page (with videos and some supplemental material) can be found here:
http://www.msri.org/workshops/689
The page for this particular talk is:
http://www.msri.org/workshops/689/schedules/18216

This is a gentle introduction to curves and more specifically algebraic curves. We look at historical aspects of curves, going back to the ancient Greeks, then ...

This is a gentle introduction to curves and more specifically algebraic curves. We look at historical aspects of curves, going back to the ancient Greeks, then on the 17th century work of Descartes.
We point out some of the difficulties with Jordan's notion of curve, and move to the polynumber approach to algebraic curves.
The aim is to set the stage to generalize the algebraic calculus of the previous few lectures to algebraic curves.
This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. The full playlist is at http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=view_all
A screenshot PDF which includes MathFoundations46 to 79 can be found at my WildEgg website here: http://www.wildegg.com/store/p101/product-Math-Foundations-screenshot-pdf

This is a gentle introduction to curves and more specifically algebraic curves. We look at historical aspects of curves, going back to the ancient Greeks, then on the 17th century work of Descartes.
We point out some of the difficulties with Jordan's notion of curve, and move to the polynumber approach to algebraic curves.
The aim is to set the stage to generalize the algebraic calculus of the previous few lectures to algebraic curves.
This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. The full playlist is at http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=view_all
A screenshot PDF which includes MathFoundations46 to 79 can be found at my WildEgg website here: http://www.wildegg.com/store/p101/product-Math-Foundations-screenshot-pdf

What is ALGEBRAIC GEOMETRY? What does ALGEBRAIC GEOMETRY mean? ALGEBRAIC GEOMETRY meaning

What is ALGEBRAIC GEOMETRY? What does ALGEBRAIC GEOMETRY mean? ALGEBRAIC GEOMETRY meaning - ALGEBRAIC GEOMETRY definition -ALGEBRAIC GEOMETRY explanation.
Source: Wikipedia.org article, adapted under https://creativecommons.org/licenses/by-sa/3.0/ license.
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.
The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves and quartic curves like lemniscates, and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.
Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.
In the20th century, algebraic geometry split into several subareas.
The mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an algebraically closed field.
The study of the points of an algebraic variety with coordinates in the field of the rational numbers or in a number field became arithmetic geometry (or more classically Diophantine geometry), a subfield of algebraic number theory.
The study of the real points of an algebraic variety is the subject of real algebraic geometry.
A large part of singularity theory is devoted to the singularities of algebraic varieties.
With the rise of the computers, a computational algebraic geometry area has emerged, which lies at the intersection of algebraic geometry and computer algebra. It consists essentially in developing algorithms and software for studying and finding the properties of explicitly given algebraic varieties.
Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology, differential and complex geometry. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory. Wiles's proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach.

18:57

Complex Analysis and algebraic geometry

In many ways, the modern revival of algebraic geometry resulted from investigations in com...

Complex Analysis and algebraic geometry

In many ways, the modern revival of algebraic geometry resulted from investigations in complex function theory. In this video, we show how Weierstrass's p-function can be used to relate cubic curves and 2-tori. We see in particular that cubic curves have an abelian group structure.
The material in this video and the other in the playlist can be found in Kirwan's "Complex algebraic curves".

The role of curvature in algebraic geometry: some history and current issues (Phillip Griffiths)

The role of curvature in algebraic geometry: some history and current issues
Plática dada por Phillip Griffiths (Institute for Advanced Study, Princeton) en el evento Algebraic Geometry in Mexico 2016 el martes 1 de noviembre del 2016.
Abstract:
Curvature has historically guided some of the most important results in algebraic geometry: vanishing theorems leading to existence results, holomorphic mappings to algebraic varieties (hyperbolicity), and global geometric applications of Hodge theory. Typically curvature methods prove a result when the variety is smooth and the curvature form is definite, and then considerable skill and technical effort extends these results to a general algebro-geometric setting when the variety is singular and the curvature is only definite on a Zariski open set. There has recently been work using curvature properties of the Hodge bundles in a family of algebraic varieties to establish results in the smooth and definite case such as the well-known conjectures “The Hodge bundle is ample on the Satake-Baily-Borel completion of a period mapping” and “The moduli space for varieties of general type is itself of log-general type”. The proofs of these results in general require non-trivial technical extensions in the directions suggested by curvature considerations, including extending the methods to include when the metric and its curvature become singular. In this talk I will briefly recall some of the classical results and then discuss some more recent results and issues just mentioned.
Video de la plática
https://youtu.be/ceBhyAeKRrs
Foto de miniatura del video
https://goo.gl/photos/A5ZVZ2sGxFpqek9x9
Página de Phillip Griffiths en Wikipedia
https://en.wikipedia.org/wiki/Phillip_Griffiths
Página de Phillip Griffiths en Phillip Griffiths en Institute for Advanced Study, Princeton
https://www.ias.edu/scholars/griffiths
Otras pláticas del evento Algebraic Geometry in Mexico 2016
https://www.youtube.com/playlist?list=PLrg-5oUhFeioieJgNVvON2WGwIh07bh8M
Otras pláticas del evento Algebraic Geometry in Mexico 2015
https://www.youtube.com/playlist?list=PLrg-5oUhFeipLL7JxLlnij0z1EKQ1dr6i
Videos Académicos de matemáticas (pláticas grabadas)
https://www.youtube.com/playlist?list=PLrg-5oUhFeiqvXI_Bxb5iOH25gXprez6o
Videos Académicos publicados
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Videos Académicos publicados en 2016
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Album de fotos miniatura de los videos Académicos publicados en 2016
https://goo.gl/photos/Wk45yoGqPe5hKTw77
Videos Académicos publicados en 2015
https://www.youtube.com/playlist?list=PLrg-5oUhFeiosqyOgMN2yPybqXIrPAhbi
Videos Académicos publicados en 2014
https://www.youtube.com/playlist?list=PLrg-5oUhFeiomqff3PibNQMJJngwB-wL6
Videos Académicos publicados en 2013
https://www.youtube.com/playlist?list=PLrg-5oUhFeipDUZKuJZM0QJDP22Cogkot
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Affine algebraic geometry: Algebraic Incarnation of Points

This is the first in a series of videos introducing affine algebraic geometry. One of the principles of algebraic geometry is that the co-ordinate ring captures all the geometry. In this video, we show how points of affine varieties can be described purely algebraically in terms of the co-ordinate ring. Hence the co-ordinate ring recovers the underlying set structure of an affine variety.
The material in this video and the entire playlist is standard and can be found in many introductory texts on algebraic geometry including Shafarevich's "BasicAlgebraic Geometry" and Hartshorne's "Algebraic Geometry".

Algebra & Geometry 1 (GCSE Higher Maths) - Exam Qs 26

This video is a tutorial on Algebra & GeometryQuestions. You should have already watched the Algebra & Geometry 1 Tutorial 25. This video is for students attempting the Higher paper AQAUnit 3 MathsGCSE, who have previously sat the foundation paper. Explanations are aimed at being as simple as possible and so students who previously did the Foundation paper can access. They would also be useful for students who have always sat the higher paper. www.hegartymaths.com http://www.hegartymaths.com/

Columbia State Community College will host summer camps for middle school students on the WilliamsonCampus during the months of June and July. The goal of each camp is to increase students’ understanding and encourage excitement for science, technology, engineering and mathematics ...BeginnerLegoRobots — June 5-8 ... In this camp, math will be introduced in creative ways – addressing topic areas such as algebra, geometry and trigonometry....

Euclidean and Algebraic Geometry, David Cox [2014]

slides for this talk: http://helper.ipam.ucla.edu/publications/ccgws1/ccgws1_11873.pdfDavid CoxAmherst College
This talk will survey some examples, mostly geometric questions about Euclidean space, where the methods of algebraic geometry can offer some insight. I will also discuss some of the limitations of this approach.
http://www.ipam.ucla.edu/abstract/?tid=11873&pcode=CCGWS1

From the MSRI introductory workshop on "Algebraic Topology" that took place at MSRI, Berkeley in January 2014. The workshop page (with videos and references and exercises) can be found here:
http://www.msri.org/workshops/685
The page for this particular talk is:
http://www.msri.org/workshops/685/schedules/17910

32:05

Algebra & Geometry 1 (GCSE Higher Maths) - Exam Qs 26

This video is a tutorial on Algebra & Geometry Questions. You should have already watched...

Algebra & Geometry 1 (GCSE Higher Maths) - Exam Qs 26

This video is a tutorial on Algebra & GeometryQuestions. You should have already watched the Algebra & Geometry 1 Tutorial 25. This video is for students attempting the Higher paper AQAUnit 3 MathsGCSE, who have previously sat the foundation paper. Explanations are aimed at being as simple as possible and so students who previously did the Foundation paper can access. They would also be useful for students who have always sat the higher paper. www.hegartymaths.com http://www.hegartymaths.com/

52:05

08. Algebraic geometry - Introducing schemes (Anton Mosunov)

Algebraic geometry seminar
Department of Pure Mathematics
University of Waterloo
November ...

Abstract:
We will briefly recall the general philosophy of non-commutative (and derived) algebraic geometry in order to establish a precise link between dg-derived category of singularities of Landau-Ginzburg models and vanishing cohomology, over an arbitrary henselian trait. We will then focus on a trace formula for dg-categories and a recent application to Bloch’s conductor conjecture. This second, and main part of the talk refers to work in progress, joint with B. Toën.

An accessible lecture by Jacob Lurie from about 2005.
The motivations behind derived algebraic geometry are presented, starting with Bezout's theorem for intersections.
A generalization of algebraic geometry using derived (topological) coordinate rings -- where identifications (quotients by ideals of defining equations) are carried out "topologically" -- allows for the proper handling of non-transversal intersections, justifying Serre's otherwise mysterious formula.
See also the paper:
http://www.math.harvard.edu/~lurie/papers/DAG.pdf
And the book:
http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf

Columbia State Community College will host summer camps for middle school students on the WilliamsonCampus during the months of June and July. The goal of each camp is to increase students’ understanding and encourage excitement for science, technology, engineering and mathematics ...BeginnerLegoRobots — June 5-8 ... In this camp, math will be introduced in creative ways – addressing topic areas such as algebra, geometry and trigonometry....

SCHOOLKIDS all over the country will be preparing for their GCSE maths exam ... Getty - Contributor ... Over the three papers, pupils will be tested on geometry, number, ratio, proportion and rates of change, algebra and statistics and probability ... Getty - Contributor ... These include. ... ....

If you've been paying attention to Imagination's recent demos you may have seen that a certain gnome has made a regular appearance on our booth ... How was the Gnome created? ... Once that was done, a UV texture map was applied to the model so the 3D geometry could be unwrapped and flattened to a set of 2D UV texture coordinates. This enables the artist to project detail onto the 2D image which corresponds with the geometry on the 3D model....

ButteSchool District #1 held its ninth annual City Hall of Fame competition on May 3 in the Butte HighAuditorium... They also had to read a novel, do algebraic equations, and correct grammatical errors in sentences. Students studied on their own time outside of school for the event ... 1 ... ....

Mason will walk across the stage tonight as a GrayElementary graduate. While many students anticipated vacations and time away from school, Mason looked forward Wednesday to spending his summer learning. “I need to keep myself in the school groove,” he said ... I didn’t get the score I necessarily wanted ... Mason researched algebra, calculus and trigonometry as an eighth-grader at Gray ... “Things like that ... .......

Starting high school could be a scary and nervous experience, but not for the incoming ninth-graders at Davis Renov Stahler YeshivaHigh School for Boys in Woodmere ...The boys heard from RabbiYisroel Kaminetsky, and General Studies principal, Dr ... The students took placement exams in algebra and the Hebrew language to determine the appropriate classes for next school year, and then were rewarded with ice cream sundaes ... ....